MATH 281A COURSE INFORMATION  


GRADING: The grade will be determined based on the homework and the take-home final.

HOMEWORKS:
(1) Homeworks will be assigned/collected roughly bi-weekly, for each homework the due date will be specified.
(2) Important: It is crucial that you learn to write the arguments briefly but correctly and completely, and that you will carefully chose what to write. For this reason I will always stipulate the maximum length of the text you may use. If you are typing, the size of the font must not be less than 10pt. I will not read any text that exceeds the specified maximum length.

MATERIAL COVERED IN MATH 281A
The material covered in Math 281A includes the following topics in descriptive set theory.
(1) Rank analysis of well-founded trees.
(2) Approximation theorem for measures.
(3) Lightface projective hierarchy and tree representations.
(4) Prewellorderings, scales and uniformization.
(5) Infinite games and Determinacy, with the following topics:
     (a) Ultrafilters under Determinacy.
     (b) Construction of norms.

MATERIAL COVERED IN MATH 281B
(6) Infinite games and determinacy, with the following topics:
     (a) Construction of scales.
     (b) Regularity properties.
     (c) Martin (cone) measure and ultrafilters below Theta.
     (d) Coding Lemma and some of its consequences.
(7) Transitive models of set theory, absoluteness, reflection theorems, inner model criterion.
(8) Connection between the Le'vy hierarchy and the projective hierarchy.
(9) Relative consistency results: background and methodology.
(10) Ordinal definability, the inner model HOD and relative consistency of the Axiom of Choice.
(11) The constructible universe.

PLAN FOR MATH 281C
(12) Constructible universe: \Delta^1_2 couterexamples for regularity properties in L, GCH in L, Mansfield theorem, the set C_2
(13) Constructible universe: Jensen hierarchy, rudimentary functions, Condensation lemma, \Sigma_1-satisfaction relation,
\Sigma_1-Skolem Function, acceptability. (14) Martin's Axiom and its basic applications.
(15) Elements of Forcing, Forcing relation, Forcing theorem
(16) Properties of forcing posets and their inluence on generic extensions: chain conditions, closure, strategic closure, distributivity
(17) Basic examples of forcing: Cohen sets, collapse forcing, Le'vy collapse, forcing a square sequence/non-reflecting stationary set, club shooting, tree forcings.
(18) Regular embeddings, dense embeddings, automorphisms, projections.
(19) Product forcing
(20) WILL BE SPECIFIED LATER

Recomended literature:
For topics (1) -- (6)
Moschovakis, Y.: Descriptive Set Theory.
Kechris, A.: Classical Descriptive set theory.
Kanamori, A.: The Higher Infinite.
For topics (7) -- (19)
Jech, T.: Set theory. Millenium edition.
Kanamori, A.: The Higher Infinite.
Kunen, K.: Set theory. An introduction to independence proofs.

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